Another way to capture the covariance structure of the observed variables is to ignore the counting nature of the data and use Gaussian Markov random fields (GMRF) [1]} to directly estimate the covariance, or Gaussian Graphical Models (GGM) [2]} to enforce sparsity on the inverse of the covariance estimate.
Unlike our proposed multinomial-Gaussian model the GMRF and GGM do not incorporate a low rank latent structure on the covariance nor do they account for the counting nature of the data.
There have been extensions of the GGM to handle multinomial observations using copulas [3]} that have been applied to microbiome analysis [4]}, [5]}, [6]}.
